Publications and other reference materials referred to herein, including reference cited therein, are incorporated herein by reference in their entirety and are numerically referenced in the following text and respectively grouped in the appended Bibliography which immediately precedes the claims.
Many applications would benefit from a fast and robust technique for acquiring the amplitude and phase spectrum of a medium. In addition, in many industrial applications, it is imperative to know the refractive-index dependence of an element or device on electromagnetic frequency, i.e. the dispersion. There is a growing demand to accurately analyze the linear and nonlinear spectra of bulk and composite materials (e.g. absorption, refractive index, Rayleigh, fluorescence, Raman, etc.), both in the areas of engineering as well as in the life sciences.
The Kramers-Kronig (KK) relations were derived in 1926/7 [Refs. 1, 2] to calculate the refractive index of a medium by its absorption measurements i.e., by a transmission or reflection spectrum measurement. Although the KK relations are merely a mathematical derivation, they have led to profound scientific capabilities, since they bridge two physical parameters that require different experimental methodologies when measured separately. Since it is usually much simpler to measure the absorption coefficient than the refractive index, mathematical relations like the KK ones, which relate the refractive index to the absorption coefficients are very appealing [Ref. 3].
In fact, the KK relations are only a special case of a very broad family of Hilbert transforms. The Hilbert transform connects the real and imaginary parts of the Fourier transform of any causal square-integrable L2 function α(t). If α(t) is real then the Hilbert transform can be rewritten (see, for example, Ref. 3)
                              Re          ⁢                      {                          a              ⁡                              (                ω                )                                      }                          =                              2            π                    ⁢          P          ⁢                                    ∫              0              ∞                        ⁢                                                                                ω                    ′                                    ⁢                  Im                  ⁢                                      {                                          a                      ⁡                                              (                                                  ω                          ′                                                )                                                              }                                                                                        ω                    ′2                                    -                                      ω                    2                                                              ⁢                              ⅆ                                  ω                  ′                                                                                        (                  1          ⁢          a                )                                          Im          ⁢                      {                          a              ⁡                              (                ω                )                                      }                          =                              -                                          2                ⁢                ω                            π                                ⁢          P          ⁢                                    ∫              0              ∞                        ⁢                                                            Re                  ⁢                                      {                                          a                      ⁡                                              (                                                  ω                          ′                                                )                                                              }                                                                                        ω                    ′2                                    -                                      ω                    2                                                              ⁢                              ⅆ                                  ω                  ′                                                                                        (                  1          ⁢          b                )            where P stands for Cauchy's principal value.
Therefore, the KK relations can be used to reconstruct the spectral transfer function H(ω)=|H(ω)|eiθ(ω) of a medium by measuring only the amplitude of the spectrum |H(ω)|, i.e., instead of applying the KK relations to the complex refractive index, one can implement it on the logarithm of the complex transfer function:ln [H(ω)]=ln |H(ω)|+iθ(ω)  (2)The KK relations can then be rewritten [3]
                                                    H            ⁡                          (                              ω                ′                            )                                                =                              2            π                    ⁢          P          ⁢                                    ∫              0              ∞                        ⁢                                                                                ω                    ′                                    ⁢                                      θ                    ⁡                                          (                                              ω                        ′                                            )                                                                                                            ω                    ′2                                    -                                      ω                    2                                                              ⁢                              ⅆ                                  ω                  ′                                                                                        (        3        )                                          θ          ⁡                      (            ω            )                          =                              -                                          2                ⁢                ω                            π                                ⁢          P          ⁢                                    ∫              0              ∞                        ⁢                                                            ln                  ⁢                                                                                H                      ⁡                                              (                                                  ω                          ′                                                )                                                                                                                                                      ω                    ′2                                    -                                      ω                    2                                                              ⁢                              ⅆ                                  ω                  ′                                                                                        (        4        )            
Recently, it has been suggested by the inventors [Ref. 4] to use the KK relation to reconstruct the impulse response of a diffusive medium, i.e. a medium which scatters the radiation such as biological tissue, clouds, smoke, cloth, etc. Such reconstructions are important in diffusive medium imaging, since they can allow, for example, distinguishing between the ballistic i.e. unscattered or ‘first’ light, the quasi-ballistic, and the diffusive light. Applications for the detection and imaging of a material's composition within a diffuse media include detection of tumors or build-up of plaque within tissue, or objects hidden by cloud cover, smoke or cloth. Clearly, the diffusive light is responsible for the blurring of the image [Refs. 5-9]. The main advantage of the approach proposed in [Ref. 4] is that the amplitude of the transfer function is easily measured, while phase measurements can be more complicated and time consuming. Therefore, the amplitude |H(ω)| is measured in a finite spectral range ωL≦ω≦ωH (actually, its square is measured, i.e., the intensity I(ω)=|H(ω)|2) and the phase is reconstructed accordingly [Ref. 10]
                                                        θ              KK                        ⁡                          (              ω              )                                =                                    -                                                2                  ⁢                  ω                                π                                      ⁢            P            ⁢                                          ∫                                  ω                  L                                                  ω                  ll                                            ⁢                                                                    ln                    ⁢                                                                                        H                        ⁡                                                  (                                                      ω                            ′                                                    )                                                                                                                                                                      ω                      ′2                                        -                                          ω                      2                                                                      ⁢                                  ⅆ                                      ω                    ′                                                                                      ,                                  ⁢                              or            ⁢                                                  ⁢                                          θ                KK                            ⁡                              (                ω                )                                              =                                    -                              ω                π                                      ⁢            P            ⁢                                          ∫                                  ω                  L                                                  ω                  ll                                            ⁢                                                                    ln                    ⁢                                                                                        I                        ⁡                                                  (                                                      ω                            ′                                                    )                                                                                                                                                                      ω                      ′2                                        -                                          ω                      2                                                                      ⁢                                  ⅆ                                      ω                    ′                                                                                      ,                            (        5        )            so that the final reconstructed transfer function is thenHKK(ω)=|H(ω)|exp[iθKK(ω)] or HKK(ω)=√{square root over (I(ω))}exp[iθKK(ω)]  (6)
The main problem with the method proposed in [Ref. 4] is the fact that the measured spectrum is always finite, and therefore the phase reconstruction may not be accurate. In fact, the integrand in Eq. (5) decays for large ω′ and in some cases (when the spectral range is wide enough) this approximation is sufficient for the reconstruction (see, for example, Ref. 10). It should be noted that the KK relations for transfer function reconstruction converge faster than in the case of index refraction reconstruction. In the former case Eq. (4) is used, which converges as ω′−2. On the other hand, in the latter case an equation of the form (3) is used, which converges as ω′−1. However, for many applications the convergence is not fast enough, leading to large error.
To improve the convergence it was suggested to derive the phase with knowledge of specific measurements (“anchoring” points) at certain “anchoring” frequencies. By doing so the power of ω′ at the denominator of Eq. (5) increases, and the convergence is improved. This technique was developed by Bachrach and Brown [Ref. 11] and Ahrenkiel [Ref. 12] and later by Palmer, Williams, and Budde [Ref. 13] and is usually called the multiply subtractive KK relation (MSKK). Specifically, if the phase is known at Q different frequencies, i.e., θ(ω1), θ(ω2), . . . θ(ωQ) then the phase can be evaluated by the relation
                                          θ            ⁡                          (              ω              )                                ω                =                                                            θ                ⁡                                  (                                      ω                    1                                    )                                                            ω                1                                      ⁢                                                            (                                                            ω                      2                                        -                                          ω                      2                      2                                                        )                                ⁢                                  (                                                            ω                      2                                        -                                          ω                      3                      2                                                        )                                ⁢                                                                  ⁢                …                ⁢                                                                  ⁢                                  (                                                            ω                      2                                        -                                          ω                      Q                      2                                                        )                                                                              (                                                            ω                      1                      2                                        -                                          ω                      2                      2                                                        )                                ⁢                                  (                                                            ω                      1                      2                                        -                                          ω                      3                      2                                                        )                                ⁢                                                                  ⁢                …                ⁢                                                                  ⁢                                  (                                                            ω                      1                      2                                        -                                          ω                      Q                      2                                                        )                                                              +                      …            ⁢                                                  ⁢                                          θ                ⁡                                  (                                      ω                    j                                    )                                                            ω                j                                      ⁢                                                            (                                                            ω                      2                                        -                                          ω                      1                      2                                                        )                                ⁢                                                                  ⁢                …                ⁢                                                                  ⁢                                  (                                                            ω                      2                                        -                                          ω                                              j                        -                        1                                            2                                                        )                                ⁢                                  (                                                            ω                      2                                        -                                          ω                                              j                        +                        1                                            2                                                        )                                ⁢                                                                  ⁢                …                ⁢                                                                  ⁢                                  (                                                            ω                      2                                        -                                          ω                      Q                      2                                                        )                                                                              (                                                            ω                      j                      2                                        -                                          ω                      1                      2                                                        )                                ⁢                                                                  ⁢                …                ⁢                                                                  ⁢                                  (                                                            ω                      j                      2                                        -                                          ω                                              j                        -                        1                                            2                                                        )                                ⁢                                  (                                                            ω                      j                      2                                        -                                          ω                                              j                        +                        1                                            2                                                        )                                ⁢                                                                  ⁢                …                ⁢                                                                  ⁢                                  (                                                            ω                      j                      2                                        -                                          ω                      Q                      2                                                        )                                                              +                      …            ⁢                                                  ⁢                                          θ                ⁡                                  (                                      ω                    Q                                    )                                                            ω                Q                                      ⁢                                                                                (                                                                  ω                        2                                            -                                              ω                        1                        2                                                              )                                    ⁢                                      (                                                                  ω                        2                                            -                                              ω                        2                        2                                                              )                                    ⁢                                                                          ⁢                  …                  ⁢                                                                          ⁢                                      (                                                                  ω                        2                                            -                                              ω                                                  Q                          -                          1                                                2                                                              )                                                  ⁢                                                                                                                (                                                            ω                      Q                      2                                        -                                          ω                      1                      2                                                        )                                ⁢                                  (                                                            ω                      Q                      2                                        -                                          ω                      2                      2                                                        )                                ⁢                                                                  ⁢                …                ⁢                                                                  ⁢                                  (                                                            ω                      Q                      2                                        -                                          ω                                              Q                        -                        1                                            2                                                        )                                                              -                                    2              π                        ⁢                          (                                                ω                  2                                -                                  ω                  1                  2                                            )                        ⁢                          (                                                ω                  2                                -                                  ω                  2                  2                                            )                        ⁢                                                  ⁢            …            ⁢                                                  ⁢                          (                                                ω                  2                                -                                  ω                  Q                  2                                            )                        ⁢            P            ⁢                                          ∫                0                ∞                            ⁢                                                                    ln                    ⁢                                                                                        H                        ⁡                                                  (                                                      ω                            ′                                                    )                                                                                                                                                                      (                                                                        ω                          ′2                                                -                                                  ω                          2                                                                    )                                        ⁢                                          (                                                                        ω                          ′2                                                -                                                  ω                          1                          2                                                                    )                                        ⁢                                                                                  ⁢                    …                    ⁢                                                                                  ⁢                                          (                                                                        ω                          ′2                                                -                                                  ω                          Q                          2                                                                    )                                                                      ⁢                                  ⅆ                                      ω                    ′                                                                                                          (        7        )            
The first terms resemble the Legendre polynomial interpolation, and only the last term (with the integral) takes account of the function's causality. In this case the integral converges as ω′−2Q so that it will converge for a narrower spectrum. In the particular case of singly subtractive KK (SSKK) relations [Refs 3, 11], i.e., when only a single ‘anchoring’ point is known, this expression reduces to
                                          θ            ⁡                          (              ω              )                                ω                =                                            θ              ⁡                              (                                  ω                  1                                )                                                    ω              1                                -                                    2              π                        ⁢                          (                                                ω                  2                                -                                  ω                  1                  2                                            )                        ⁢            P            ⁢                                          ∫                0                ∞                            ⁢                                                                    ln                    ⁢                                                                                        H                        ⁡                                                  (                                                      ω                            ′                                                    )                                                                                                                                                                      (                                                                        ω                          ′2                                                -                                                  ω                          2                                                                    )                                        ⁢                                          (                                                                        ω                          ′2                                                -                                                  ω                          1                          2                                                                    )                                                                      ⁢                                  ⅆ                                      ω                    ′                                                                                                          (        8        )            
As was demonstrated in the literature [Ref. 3] this technique improves the reconstruction for less than infinite spectral boundaries, i.e. in the finite range ωL≦ω≦ωH. However, the main disadvantage of this technique is that in most cases there is absolutely no a-priori information on the phases, and since phase measurements are difficult to carry out it may be difficult to implement this promising technique.
The inventors have developed in the past a technique known as Spectral Ballistic Imaging (SPEBI) [Refs. 10, 14-16]. In this technique, the full (amplitude and phase) spectrum of the medium is obtained, by direct measurement of the amplitude spectrum and the derivative of the phase spectrum. The phase spectrum is then calculated by integrating the phase-derivative spectrum. The phase-derivative spectrum is much easier to measure than the actual phase spectrum, as disclosed in the prior art, since it is significantly simpler than direct phase-measuring techniques such as interferometry or ellipsometry. For example, for a source in the optical regime, it is done by modulating the light source at an RF frequency, e.g. 1 GHz, illuminating the medium, detecting the reflected or transmitted light, and then measuring the RF phase lag of the detected light with respect to the illuminating light. The phase lag divided by the modulation frequency is approximately the phase derivative at the carrier frequency of the light. By carrying out this measurement over a set of frequencies within the spectral range of interest, and then integrating the result, the phase spectrum of the medium is obtained. This gives the complete spectrum of the medium, and completely characterizes its' optical properties. If it is desired to obtain a ‘ballistic’ image of the medium, it can now be computed by carrying out an inverse Fourier transform on the complex (amplitude and phase) spectrum, to obtain the optical impulse response of the medium, as explained above.
The SPEBI technique, although proven to be very accurate, has a serious drawback: the phase derivative must be measured for many optical carrier frequencies. Depending upon the application, performing the required number of measurements can be too time-consuming. On the other hand, the KK technique relies on amplitude measurements which are inherently fast, and it has been shown above in the review of the prior art that it can be used to derive the impulse response of a diffuse medium, however it may be less accurate than SPEBI. As described above, phase-anchoring techniques can be employed to improve the accuracy of KK, but phase measurements are difficult and susceptible to noise.
Another technique known to be useful for determining the phase spectrum from amplitude measurements is based upon the maximum entropy model (MEM). The MEM technique is described in Ref. [3] and other references therein. However, as in the case of the KK method, its' implementation for diffuse media has not been disclosed or demonstrated in the past. The power spectrum |H(ω)|2 is measured in a finite spectral range ωL≦ω≦ωH, and then this spectrum is fitted to the MEM model, given by
                                                                    H              ⁡                              (                v                )                                                          2                =                                                          β                                      2                                                                                              A                  M                                ⁡                                  (                  v                  )                                                                    2                                              (        9        )            where AM(ν)=1+Σm=1Mαmexp(−i2πmν) is a MEM polynomial given by the MEM coefficients αm and by the normalized frequency ν≡(ω−ωL)/(ωH−ωL). The unknown coefficients αm and |β|2 are obtained by the system of equations:
                                                        (                                                                                          C                      ⁡                                              (                        0                        )                                                                                                                        C                      ⁡                                              (                                                  -                          1                                                )                                                                                                  ⋯                                                                              C                      ⁡                                              (                                                  -                          M                                                )                                                                                                                                                        C                      ⁡                                              (                        1                        )                                                                                                                        C                      ⁡                                              (                        0                        )                                                                                                  ⋯                                                                              C                      ⁡                                              (                                                  1                          -                          M                                                )                                                                                                                                  ⋮                                                        ⋮                                                        ⋱                                                        ⋮                                                                                                              C                      ⁡                                              (                        M                        )                                                                                                                        C                      ⁡                                              (                                                  M                          -                          1                                                )                                                                                                  ⋯                                                                              C                      ⁡                                              (                        0                        )                                                                                                        )                        ⁢                          (                                                                    1                                                                                                              a                      1                                                                                                            ⋮                                                                                                              a                      M                                                                                  )                                =                      (                                                                                                                            β                                                              2                                                                                                0                                                                              ⋮                                                                              0                                                      )                          ⁢                                  ⁢                              where            ⁢                                                  ⁢                          C              ⁡                              (                t                )                                              =                                    ∫              0              1                        ⁢                                                                                                  H                    ⁡                                          (                      v                      )                                                                                        2                            ⁢                              exp                ⁡                                  (                                      ⅈ2π                    ⁢                                                                                  ⁢                    tv                                    )                                            ⁢                              ⅆ                v                                                                        (        10        )            is an autocorrelation function. As in the KK technique, it is desired to calculate the true complex spectrum
                              H          ⁡                      (            v            )                          =                                                                          H                ⁡                                  (                  v                  )                                                                    ⁢                          exp              ⁡                              (                                  ⅈθ                  ⁡                                      (                    v                    )                                                  )                                              =                                                                                        β                                                  ⁢                                  exp                  ⁡                                      (                                          ⅈθ                      ⁡                                              (                        v                        )                                                              )                                                                                                                                    A                    M                                    ⁡                                      (                    v                    )                                                                                        .                                              (        11        )            
However, due to inaccuracies inherent in the MEM technique, the complex polynomial AM(ν) will have an error in the phase spectrum, i.e. AM(ν)=|AM(ν)|exp(iψ(ν)) instead of the true AM(ν)=|AM(ν)|exp(−iθ(ν)), Therefore, the error phase spectrumφ(ν)=θ(ν)−ψ(ν)  (12)must be determined in order to determine the true spectrum θ(ν).